#386471

order	512
nilpotency class	3
isoclinism	isoclinic class of #30490
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2366
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1011 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,7)(4,6)(5,8)(9,10,15,20,16,22,14,21)(11,23,26,17,27,34,32,35)(12,18,29,37,30,19,13,31)(24,38,28,33,39,36,25,40);(9,11)(10,17)(12,28)(13,24)(14,32)(15,26)(16,27)(18,36)(19,38)(20,34)(21,23)(22,35)(25,30)(29,39)(31,33)(37,40);(1,3,4,8)(2,5,6,7)(9,12,14,13,16,30,15,29)(10,18,21,31,22,19,20,37)(11,24,26,28,27,39,32,25)(17,33,34,36,35,40,23,38);(2,6)(5,7)(9,13,14,30,16,29,15,12)(10,19,20,31,22,18,21,37)(11,25,26,24,27,28,32,39)(17,33,23,38,35,40,34,36)

Polycyclic presentation:

2^9|a^2=h,c^2=fg,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293667);